Lesson 3 2 Differentiability AP Calculus Mrs Mongold

Lesson 3 -2: Differentiability AP Calculus Mrs. Mongold

To be differentiable, a function must be continuous and smooth. Derivatives will fail to exist at: corner cusp vertical tangent discontinuity

Derivatives Don’t Exist at… • Corners: LHD ≠ RHD • Cusp: LHD ≠ RHD or in extreme cases where the slopes of the lines approach ±∞ • Vertical Tangent, where slopes approach ±∞ from both sides • Discontinuity: when one or both of the one-sided derivatives do not exist

Corner • f(x)=|x|

Cusp • f(x) =

Vertical Tangent • f(x) =

Discontinuity

Most of the functions we study in calculus will be differentiable.

Example • Find where the function is not differentiable – Find all points in the domain of f(x)=|x – 2|+3 where f is not differentiable. • Think Graphically Absolute Values have corners so what is the corner point?

Differentiability Implies Local Linearity • This means that the function at point a closely resembles it’s own tangent like very close to a. • Curves will “straighten out” when we zoom in on them at a point of differentiability

Example • Look at f(x)=xcos(3 x) – Window #1 [-4, 4] by [-3, 3] – Window #2 [1. 7, 2. 3] by [1. 7, 2. 1] – Window #3 [1. 93, 2. 07] by [ 1. 85, 1. 95]

Homework • Use your calculator answer if either of these functions are differentiable at x = 0 • f(x) = |x| +1 and g(x) = • Book page 111/1 -4 and 11 -16